this paper, discrete-time sliding mode control is proposed for implementing an induction motor drive without varying the system structure. Very simple constant.
IEEE Transactions on Energy Conversion, Vol. 9, No. 4, December 1994
645
A FMED STRUCTUREDISCRETE-TIME SLIDING MODE CONTROLLER FOR INDUCTION MOTOR DRIVES
Ching-Tsai Pan*, Ting-Yu Chang* and Chin-Ming Hong** *DEPARTMENTOF ELECTRICALENGINEERING, NATIONAL TSING HUA UNIV., TAIWAN,R.O.C. **DEPARTMENTOF INDUSTRIAL EDUCATION, NATIONAL TAIWAN NORMAL UNIV., TAIWAN,R.O.C. --In this paper, discrete-time sliding mode control is proposed for implementing an induction motor drive without varying the system structure. Very simple constant gain reaching control and sliding control are obtained to eliminate the chattering phenomenon and steady state error as well as to achieve robustness and minimized computation. For integration of the whole system, a simple field acceleration control algorithm is also derived to coordinate with the sliding mode control such that a simple 8 bit microcomputer can be used to implement a high performance induction motor drive.
Kevwords: fixed structure, sliding mode control, induction motor drive INTRODUCTION Due to advantages of induction motors, such as ruggedness, high reliability, low cost and minimum maintenance, induction motor drives are gradually replacing DC motor drives. However, for high performance induction motor drives, implementation of the conventional vector control involves a lot of computations. Hence, a high speed microcomputer or multiprocessor [1-21 is usually required to fulfill the need of computation speed. Besides, the dynamic perfommce is also influenced very much due to rotor resistance change. Hence, complicated control is required to achieve robustness .
made insensitive to plant parameter changes and external disturbances. However, up to now, most existing sliding mode controls were developed for continuous systems. Very few discrete-time variable structure controls are available in existing literature [6-81. In addition, the resulting control algorithms require lots of complex calculations. Also, the resulting controller requires an additional switch to vary the structure of the system to achieve the sliding mode control. This not only results in a chattering control which may excite unmodeled high frequency plant dynamics causing stability problems but also increases the hardware complexity.
In this paper, the authors first propose a very simple field acceleration control algorithm for the inner torque control loop of the induction motor drive to achieve linear and instantaneous torque control Without transient oscillations. Then, in the outer position control loop, a simple fixed structure discrete-time sliding mode control (DSMC) is proposed to eliminate the chattering phenomenon. The proposed sliding mode control has the merits of requiring minimum computation and without steady state error. It is seen that through the above integration, it is now possible to use a low cost 8051 microcomputer to implement a high performance induction motor drive. THE PROPOSED FIELD ACCELERATION CONTROL,
On the other hand variable structure controllers have been studied for three decades [4-91. By using the sliding mode control the dynamic behavior of a system can be 94 WM 026-5 EC A paper recommended and approved by the IEEE Electric Machinery Committee of the IEEE Power Engineering Society for presentation at the IEEE/PES 1994 Winter Meeting, New York, New York, January 30 - February 3, 1994. Manuscript submitted July 29, 1993; made available for printing January 14, 1994.
m Due to nonlinearity and high coupling of the induction motor control, a common vector controller with a synchronous rotating reference frame is usually chosen to decouple the torque control of the induction motor drives. Thus, by analogy with a separately excited dc motor control, one component of the rotor flux is forced to zero to obtain the desired stator current control. This control is then transformed back to the stationary reference frame as the
0885-8969/94/$04.00 0 1994 IEEE
Authorized licensed use limited to: National Cheng Kung University. Downloaded on June 23, 2009 at 21:12 from IEEE Xplore. Restrictions apply.
646
command signal of the induction motor stator current. However, this process requires very heavy computation burden. To minimize the computations and achieve a fully digital induction motor drive system, a simple field accelerationcontrol algorithm is derived below.
where A is defined as follows:
3I Ae
'' s i n ( w s , t ) =Im
.
. ..
CM ( R - 1 U) L ) Lr(Rr+. I w s l L r )
I,@
-
( -.) :-I
f ,
(5)
Consider the basic mathematical model of a three phase induction motor [ 11
Hence, if the stator current vector is controlled such that the following relation is satisfied
-. - -(Rr
- R , L : - R,M2 R , M L , - R , M 2 )w,M -~ L , ( L , L , - M ~ )L , ( L , L , - M ~ )L J RrM -R, IW, L, Lr
0r
I s =
MRr
+
.Iws,Lr)
(6)
+
+
Z,(k) LsLr-M2[ 0 Lr
]
(
then one can achieve instantaaeous torque response without incurring any oscillation,Le, the generated torque takes the following form
T , = -2l w ~ ( $ ~ * ? : ) (7)
3
where +
I s =id
+
ji
6, = h d r+ jh,, z
g
Thus, given a torque command T * and rotor flux magnitude O o , then one has
: stator current space vector : rotor flux space vector
( k )= u d k) + j u q c ( k ) : inverter voltage vector,
3R.T'
k =0, 1,..6
U)=U),+-
20;
: the imaginary part
Im
During a short sampling period, the rotor speed U) ,can be assumed essentially constant. Thus, corresponding to an exciting stator current vector 7, with electrical angular frequency CAIand constant amplitude, namely one can solve the second row of (1)
7 , = I Oel"'t ,
where C E
(13
- w,
: a constant decided by the system initial conditions. : the slip frequency of the induction motor.
Also by substituting 6, = + o e i e einto (6) and separating the real and imaginary p m one can obtain
3L,T' $0 I d S ( e e )= --case,- 20 M sin e e M
(9)
where 8, is the angular position of the rotor flux vector. Since the time response of a mechanical system response is much slower than the sampling time, h , of the controller,it is reasonable to use the following approximation
e e ( n + 1) = e , ( n ) +~ ) * h
(11)
By using the above solution it is straightforward to find: -Rrf
=21&rl2 w s I+ A e e 3 Rr
Lr
Therefore the desired field acceleration control algorithm can be put into the following form
-2Rp
C 2 W s el sin(cu,,t) + -
T
1
Rr
(4)
Authorized licensed use limited to: National Cheng Kung University. Downloaded on June 23, 2009 at 21:12 from IEEE Xplore. Restrictions apply.
647 B J A
: coefficient of friction : moment of inertia :a factor of the drive system
T * ( k ) : torque command of the drive system
Z,,[n+ 1 ] =
3 L T (n)
$0
2$0M
M
cosO,(n+ I)+-slnO,(n+
1)
F
:thedisturbance
To achieve a compact solution, the switching hyperplane of the proposed sliding mode control is intentionally chosen as Thus, one can see that by using the above algorithm, it is not necessary to calculate the time varying coordinate transformation. Also the torque response can be considered as instantaneous without oscillation. Hence, for later outer loop control, the torque control subsystem can simply be considered as a power amplifier with gain K
a(k)-Cx(k)+e,(k)=O
(17)
Thus, from 16) and (17) one can obtain the sliding mode dynamics:
f.
o(k+l)=CD
T
I
T - I Et
The proposed configuration of the DSMC combines an integral control as shown in Fig.1 where D , and D , are constants to be determined from later controller design.
Now let the control T ' ( k ) be decomposed into two parts as follows: T ' ( k ) = T l ( k ) +T F ( k )
Define the position error e , and its derivative e as follows:
(19)
where the reaching control T : ( k) is used to force the states to reach the predetermined hyperplane and the sliding control T f ( k ) is used to maintain the states on the switching hyperplane. First assume the external disturbance F = Oand let u(k + 1 ) = o ( k ) to derive the slidingcontrol :
Then, the dynamical model of the drive system can be described as follows
Next consider the reaching control. To eliminate the chattering phenomenon, the authors propose the following constraint where H is the sampling time of the proposed DSMC and x ( k ) is a state variable as shown in Fig. 1,and PROPOSED DSMC
to keep the system states on the same side of the predetermined hyperplane. Also, to guarantee stability of the sliding mode control, another constraint is imposed
[ ~ ( k 1+) - ~ ( k ) ] ~ < (0 k )
Fig. 1 Block diagram of the proposed discrete-time sliding mode control system.
(22)
Hence, substituting (18), (19) and (20) into (21) and (22) yields the following constraints
Authorized licensed use limited to: National Cheng Kung University. Downloaded on June 23, 2009 at 21:12 from IEEE Xplore. Restrictions apply.
648
b
K,AH T:(k)a(k)>0 .J
n'=O
From constraints (23) and (24) one can see that a simple expression O
